9 problems

Let $(X_{\mathbb C},L_{\mathbb C})$ be a polarized smooth complex projective variety. A test configuration and K-polystability are understood in the Donaldson–Futaki sense: the Don…

Let $M$ be a closed hypersurface with constant mean curvature and constant scalar curvature in the unit sphere $\mathbb{S}^{n + 1}$. Generalized Chern conjecture. For each $n$ and…

Let $(M,\omega,J)$ be a compact Kähler manifold, and let $K$ be a maximal compact subgroup of $\operatorname{Aut}_0(M)$. Let ${\mathcal H}$ be the space of Kähler potentials and le…

Let $(M,[\omega],J)$ be a compact Kähler manifold, let … and let $\mathcal K$ be the K-energy on Kähler potentials. A $C^{1,\bar 1}$ potential is a weak Kähler potential whose comp…

Torus-relative stability conjecture. If

A CPE metric is a 3-tuple $(M^n,g,f)$, where $(M^n,g)$ is a compact oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature, and $f$ is a smooth function…

Let $(M,\xi)$ be a Sasakian manifold polarized by the Reeb vector field $\xi$, with polarized affine cone $(Y,\xi)$, where $Y=C(M)$ is the fixed complex cone. A constant scalar cur…

Donald's geodesic stability conjecture. The following are equivalent:

Let $(X,J)$ be a polarized manifold, with $L$ an ample line bundle on $(X,J)$. A polarization $(X,J,L)$ is K-stable in the sense of algebraic K-stability, and $c_1(L)$ denotes the…